Problem: Simplify and expand the following expression: $ \dfrac{4p}{3p + 3}+\dfrac{p + 2}{p + 10} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(3p + 3)(p + 10)$ Multiply the first term by $\dfrac{p + 10}{p + 10}$ $ \begin{align*} \dfrac{4p}{3p + 3} \times \dfrac{p + 10}{p + 10} & = \dfrac{(4p)(p + 10)}{(3p + 3)(p + 10)} \\ & = \dfrac{4p^2 + 40p}{(3p + 3)(p + 10)}\end{align*} $ Multiply the second term by $\dfrac{3p + 3}{3p + 3}$ $ \begin{align*} \dfrac{p + 2}{p + 10} \times \dfrac{3p + 3}{3p + 3} & = \dfrac{(p + 2)(3p + 3)}{(p + 10)(3p + 3)} \\ & = \dfrac{3p^2 + 9p + 6}{(p + 10)(3p + 3)}\end{align*} $ Now we have: $ = \dfrac{4p^2 + 40p}{(3p + 3)(p + 10)} + \dfrac{3p^2 + 9p + 6}{(p + 10)(3p + 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4p^2 + 40p + 3p^2 + 9p + 6}{(3p + 3)(p + 10)} $ $ = \dfrac{7p^2 + 49p + 6}{(3p + 3)(p + 10)}$ Expand the denominator: $ = \dfrac{7p^2 + 49p + 6}{3p^2 + 33p + 30}$